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(B) The time period $T$ of a charged particle moving in a uniform magnetic field $B$ is given by the formula $T = \frac{2\pi m}{qB}$, where $m$ is the

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$1: 2$

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(B) The time period $T$ of a charged particle moving in a uniform magnetic field $B$ is given by the formula $T = \frac{2\pi m}{qB}$, where $m$ is the mass and $q$ is the charge of the particle.For a proton, $m_p = m$ and $q_p = e$. Thus, $T_p = \frac{2\pi m}{eB}$.For an alpha particle, $m_{\alpha} = 4m$ and $q_{\alpha} = 2e$. Thus, $T_{\alpha} = \frac{2\pi (4m)}{(2e)B} = \frac{4\pi m}{eB}$.The ratio of the time taken by the proton to the time taken by the alpha particle is $\frac{T_p}{T_{\alpha}} = \frac{2\pi m / eB}{4\pi m / eB} = \frac{2}{4} = \frac{1}{2}$.Therefore, the ratio is $1: 2$.

$A$ proton and an alpha particle moving with equal speeds enter normally into a uniform magnetic field. The ratio of times taken by the proton and the alpha particle to make one complete revolution in the magnetic field is

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